Mixed monotone decomposition of dynamical systems with application |
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Authors: | Tianguang Chu Lin Huang |
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Institution: | (1) Department of Mechanics and Engineering Science, Peking University, 100871 Beijing, China |
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Abstract: | The notions of mixed monotone decomposition of dynamical systems are introduced. The fundamental idea is to make an elaborate
use of the natural growth and decay mechanism inherent in the structure of a dynamical systems to decompose its dynamics into
increase and decrease parts, and thereby to constitute an augmented dynamical system as the secalled “two-sided comparison
system” of the original one. The corresponding two-sided comparison theorems show that the solution of the comparison system
gives lower and upper bounds of that of the original system. Therefore, the properties of a dynamical system can be obtained
through the study of its two-sided comparison system. Compared with the conventional comparison method in literature, the
mixed monotone decomposition method developed herein takes in a natural way structural properties of dynamical systems into
account instead of requiring strict conditions of (quasi-)monotonicity on them, and could yields refined, usually nonsymmetrical,
state estimates, and thus is more suitable for systems with nonsymmetrical state constraints. As an application of the method,
a sufficient condition is established for the global asymptotic stability of the trivial solution of a class of continuous-time
systems with nonsymmetrical state saturation. The condition is given in terms of coefficients and state saturation levels
of such systems, and contains as a special case a recent result on systems with symmetric state saturation in literature. |
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Keywords: | lower and upper mixed (quasi-)monotone decomposition two-sided comparison theorems dynamical systems nonsymmetrical state saturation global asymptotic stability |
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